Quadratic Maps and Bockstein Closed Group Extensions

Abstract

We study central extensions E of elementary abelian 2-groups by elementary abelian 2-groups. Associated to such an extension is a quadratic map which determines the extension uniquely. The components of the map determine a quadratic ideal in a polynomial algebra and we say that the extension is Bockstein closed if this ideal is invariant under the Bockstein operator. We find a direct condition on the quadratic map that characterizes when the extension is Bockstein closed. Using this we show for example that quadratic maps induced from the fundamental quadratic map on gl_n, Q(A)=A+A^2 yield Bockstein closed extensions. We also show that this condition is equivalent to a certain liftability of the extension and under certain conditions, to the fact that Q corresponds to a 2-power map of a restricted 2-Lie algebra. For the class of 2-groups coming from these type of extensions, under a 2-power exact condition we also compute the mod 2 cohomology ring of the corresponding groups - this includes the upper triangular congruence subgroups among other examples.